**Structural Lattice Reduction: Generalized Worst-Case to Average-Case Reductions**

*Nicolas Gama and Malika Izabachene and Phong Q. Nguyen and Xiang Xie*

**Abstract: **In lattice cryptography, worst-case to average-case reductions rely on two problems: Ajtai's SIS and Regev's LWE,
which refer to a very small class of random lattices related to the group G=Z_q^n.
We generalize worst-case to average-case reductions to (almost) all integer lattices,
by allowing G to be any (sufficiently large) finite abelian group.
In particular, we obtain a partition of the set of full-rank integer lattices of large volume
such that finding short vectors in a lattice chosen uniformly at random from any of the partition cells is as hard as finding short vectors in any integer lattice.
Our main tool is a novel group generalization of lattice reduction, which we call structural lattice reduction: given a finite abelian group $G$ and a lattice $L$,
it finds a short basis of some lattice $\bar{L}$ such that $L \subseteq \bar{L}$ and $\bar{L}/L \simeq G$.
Our group generalizations of SIS and LWE allow us to abstract lattice cryptography, yet preserve worst-case assumptions.

**Category / Keywords: **foundations / Lattices, worst-case to average-case reductions, SIS, LWE

**Date: **received 23 Apr 2014

**Contact author: **pnguyen at di ens fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20140424:223558 (All versions of this report)

**Short URL: **ia.cr/2014/283

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